Catalog Information

Title

(Credit Hours:Lecture Hours:Lab Hours)

Offered

Prerequisite

Description

Desired Learning Outcomes

Prerequisites

Students to have mastered Math 313 (linear algebra) and Math 371 (group theory and ring theory) prior to enrolling in Math 372.

This is a second course in abstract algebra focusing on field theory. The course is aimed at undergraduate mathematics majors, and it is strongly recommended for students intending to complete a graduate degree in mathematics. In addition to being an important branch of mathematics in its own right, abstract algebra is now an essential tool in number theory, geometry, topology, and, to a lesser extent, analysis. Outside of mathematics, algebra also has applications in cryptography, coding theory, quantum chemistry, and physics.

Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

Ring Theory

Basic Definitions

Examples of rings (both commutative and noncommutative)

Ideals and ring homomorphisms

Quotient rings

Prime and maximal ideals

Polynomial rings over fields

Factorization in polynomial rings

Irreducible polynomials

Field of fractions of a domain

Field Theory

Extensions of fields

Field extensions via quotients in polynomial rings

Automorphisms of fields

Field of characteristic 0 and prime characteristic

Galois extensions and Galois groups

The Galois correspondence

Independence of characters

Fundamental Theorem of Galois Theory

Fundamental Theorem of Algebra

Roots of unity

Solvability by radicals

Insolvability of the quintic

Textbooks

Possible textbooks for this course include (but are not limited to):

Joseph Rotman, Galois Theory (Second Edition), Springer, 1998.

David Dummit and Richard Foote, Abstract Algebra (Third Edition), Wiley, 2003. (The chapters on fields and Galois theory and probably including some of the material on rings.)

Ian Stewart, Galois Theory (Third Edition), Chapman Hall, 2004.

Additional topics

The instructor may cover additional topics beyond the minimal requirements. Possible topics include (but are not limited to): introduction to algebraic numbers, applications of field extensions to cryptography, applications of field extensions to diophantine analysis, relations of field theory to algebraic geometry, construction of algebraically closed fields using Zorn's lemma, introduction to computer calculations in abstract algebra.